on a blackboard. One operation is to delete and replace it by 
; or by 
; or by 
if 
. The final goal of Azambuja is to write the number 
after performing a finite number of operations. Show that if the initial number written is 
, then Azambuja cannot reach his goal.Do you really need a hint? Just try it yourself! [/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.23.3"]It is always a good idea to try using the invariance principle in such problems. [/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.23.3"]Make a change of variables to see patterns. [/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.23.3"]Note that the operation is restricted to rational numbers. Hence, writing $latex q=\frac{r}{s}=(r,s)$ could help. [/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.23.3"]Let us denote by $latex q_n$ the number on the board at the $latex n$th step. We shall use the new variable $latex a_n=2q_n-1$ (just to simplify the denominator). Clearly, $latex a_{n+1}$ is either $latex a_n\pm 2$ or $latex -\frac{1}{a_n}$. Writing $latex a_n=(r_n,s_n)$, this means that $latex (r_{n+1},s_{n+1})$ is either $latex (r_n \pm 2s_n,s_n)$ or $latex (-s_n,r_n)$. Thus, we need to find out if $latex (-1008,1009)$ is reachable starting from $latex (-1,1)$. However, (odd, odd) pairs can produce only other (odd,odd) pairs under this operation, and $latex (-1008,1009)$ is an (even, even) pair. Hence $latex \frac{1}{2018}$ cannot be reached starting from 0. [/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.22.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px"]
Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year.Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.[/et_pb_blurb][et_pb_button button_url="https://cheenta.com/matholympiad/" url_new_window="on" button_text="Learn More" button_alignment="center" _builder_version="3.23.3" custom_button="on" button_bg_color="#0c71c3" button_border_color="#0c71c3" button_border_radius="0px" button_font="Raleway||||||||" button_icon="%%3%%" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3" background_layout="dark"][/et_pb_button][et_pb_text _builder_version="3.22.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px"]
Publishing original scientific research in a peer-reviewed journal is an achievement many scholars work toward for years. Achieving this feat early in one's academic journey requires exceptional intellectual rigor, creativity, and persistence. Aaditya Punatar and Muskan Sadana have reached this remarkable milestone, successfully publishing their collaborative research paper in the prestigious Journal of Emerging Investigators […]
Her project on quantum-accelerated Steiner trees for 15-minute cities earned a place among the finalists at the world's largest pre-college STEM competition. Prishaa Shrimali of Marymount School of New York, was selected as a Grand Finalist at the 2026 Regeneron International Science and Engineering Fair (ISEF) in Phoenix, Arizona. Her project was Optimizing Urban Accessibility: […]
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
